Bone with TrackTo constraint isn't rotating properly in poses on 1st frame [video included]
Thanks to comments in the original post, I was able to find the problem and create a fix.

THE PROBLEM: 
In this case, the bone with the TrackTo constraint was targeting a mesh object (a cylinder) which was a child of the armature, creating a circular dependency and producing the error I was experiencing.

THE FIX: 
Positioned in the same place as the cylinder that was being targeted is another bone controlling the upper part of the excavator mechanism. I changed the target of the TrackTo constraint to that bone above it, thus eliminating the circular dependency. Now, even from frame one, the rotation of the bone with the TrackTo constraint is right where it is supposed to be.

Thanks again to everyone for their help!